Mathematical biology · physiological flows · evolving networks
Coupled models of flow, transported signals, mechanics, and geometry evolution in tissue scaffolds and endothelial confinement.
These projects ask how biological function emerges inside narrow, deformable, and often networked microstructures. In both settings, fluid carries nutrients or chemical signals; cells respond to those fields; mechanics changes the available space; and the altered geometry redistributes the flow and transport.
Biological tissues rarely behave as passive containers. Their internal passages carry fluid and dissolved signals, while cells consume nutrients, generate forces, adhere to walls, and remodel the surrounding geometry. The resulting feedback is both local and global: a change in one channel modifies its resistance, redirects flow elsewhere, and changes the biological conditions throughout the structure.
The two projects differ biologically, but they belong to the same mathematical class: transport through deformable biological conduits whose geometry evolves in response to the transported field and local mechanics.
Tissue-engineering scaffolds provide a porous framework in which nutrient-rich fluid supports cell proliferation. Existing models often resolve either one pore in detail or the scaffold only at a bulk scale. The network model fills the gap: vertices represent pore junctions, edges represent slender scaffold channels, and the full architecture redistributes pressure, nutrients, shear stress, and growth.
The central mechanical feature is a competition between two radius changes. Tissue accumulation narrows a channel, while elastic deformation of the scaffold can widen it. Because both effects alter conductance, growth and mechanics feed back into the network flow.
Hagen–Poiseuille conductances determine edge fluxes,
while nutrient concentration satisfies an edgewise advection–diffusion–reaction equation and is mixed by flux balance at network vertices.
In nondimensional form, the channel radius evolves as
The first term represents nutrient- and shear-dependent tissue growth; the second represents pressure-driven elastic expansion.
Same feedback structure, different biology
In the scaffold, biological growth gradually narrows many connected pores. At the blood–brain barrier, a moving immune cell locally softens and opens a viscoelastic endothelial cleft while squeezing through it.
Neutrophil diapedesis is the passage of an immune cell through the narrow cleft between endothelial cells. At the blood–brain barrier, this process matters for immune surveillance, neuroinflammation, and potential cell-mediated drug delivery. The challenge is irreducibly multiphysical: pressure-driven flow, chemokine transport, receptor binding, endothelial viscoelasticity, and cell motion all influence one another.
We model the cleft as a slender, fluid-filled channel with Kelvin–Voigt walls and represent the neutrophil as a moving packet. ICAM-1/\(\beta_2\)-integrin binding locally softens the endothelium; CXCR2 uptake reshapes the CXCL8 field; and the cell advances through the same lubrication resistance whether it is driven by rear pressure or front-edge chemotactic traction.
The reduced wall law couples pressure to elastic restoring stress and wall viscosity,
with the local modulus \(E(z,t)\) reduced beneath the adhered neutrophil.
The depth-averaged chemokine satisfies an advection–diffusion equation with localized CXCR2 uptake. The leading edge obeys the schematic force balance
together with a confinement threshold that can arrest motion when the gap becomes too narrow.
The scaffold model is already posed on a random network. The BBB model currently resolves one intercellular cleft, but the endothelial packing around a blood vessel naturally generates multiple adjacent and tortuous routes. A future network formulation would treat clefts as edges and junctions or route changes as vertices, allowing local chemotaxis and viscoelastic opening to interact with global path selection.
This creates a common program in mathematical biology: derive reduced edge models from continuum mechanics, close them through physically meaningful vertex conditions, and determine how biological transport and mechanics depend on network geometry.
These simulations vary the chemotactic constant \(\chi\) and the effective lumen-side driving pressure \(p_{\mathrm{eff}}\). Red identifies the chemotactic regime; blue identifies the pressure regime.
Both driving mechanisms are weak, so the neutrophil penetrates only slightly before its motion becomes very slow. The small chemotactic traction cannot overcome the strong lubrication resistance created by confinement, while the small pressure produces only modest opening of the endothelial cleft.
The stronger chemotactic traction pulls the leading edge steadily into the cleft even though the lumen-side pressure remains weak. Chemokine uptake sharpens the signal near the cell front, sustaining forward motion, while the endothelial deformation remains relatively localized and modest.
The larger pressure visibly distends the compliant endothelial cleft and reduces geometric confinement. Cell progression is nevertheless limited by the weak chemotactic pull, especially during the initial entry stage; the movie therefore shows substantial wall deformation relative to the distance travelled by the neutrophil.
The two driving mechanisms reinforce one another. Strong chemotactic traction pulls the cell front forward, while the elevated lumen-side pressure opens the cleft and supplies an additional rearward push. The result is the fastest and deepest transit of the four regimes, with a moving wall deformation that follows the neutrophil and relaxes behind it.